AOE-Trails Constructing for a Plane Connected 4-Regular Graph
نویسندگان
چکیده
The polynomial algorithm for constructing a special Eulerian cycle in a plane connected 4-regular graph G = (V,E) is presented in the paper. First of all any cycle of the passed edges does not contain any unpassed edges (the condition of ordered enclosing (OE-cycle). At second, the next edge of this cycle can be chosen from one of two neighbouring edges (left or right) of a cyclic order for the end-vertex of the current edge (the A-trail condition). The computing complexity of this algorithm is O(|E(G)| log |V (G)|).
منابع مشابه
Orthogonal A-Trails of 4-Regular Graphs Embedded in Surfaces of Low Genus
Anton Kotzig has shown that every connected 4-regular plane graph has an A-trail, that is an Euler trail in which any two consecutive edges lie on a common face boundary. We shall characterise the 4-regular plane graphs which contain two orthogonal A-trails, that is to say two A-trails for which no subtrail of length 2 appears in both A-trails. Our proof gives rise to a polynomial algorithm for...
متن کاملCircle-Representations of Simple 4-Regular Planar Graphs
Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, (a) we affi...
متن کاملOn a C4-ultrahomogeneous oriented graph
The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected ~ C4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in w...
متن کاملLovász-Plummer conjecture on Halin graphs
A Halin graph, defined by Halin [3], is a plane graph H = T ∪ C such that T is a spanning tree of H with no vertices of degree 2 where |T | ≥ 4 and C is a cycle whose vertex set is the set of leaves of T . In his work, as an example of a class of edge-minimal 3-connected plane graphs, Halin constructed this family of plane graphs, which have many interesting properties. Lovász and Plummer [5] n...
متن کاملOn a conjecture of Lovász on circle-representations of simple 4-regular planar graphs
Lovász conjectured that every connected 4-regular planar graph G admits a realization as a system of circles, i.e., it can be drawn on the plane utilizing a set of circles, such that the vertices of G correspond to the intersection and touching points of the circles and the edges of G are the arc segments among pairs of intersection and touching points of the circles. In this paper, we settle t...
متن کامل